A short note on the mystery of aerodynamic lift and its competing theories.
Last weekend, I flew to Clemson University, South Carolina to visit my friend for the July 4th weekend. Every time I fly, I cannot help but wonder how the physics happening in a thin layer of air surrounding a wing of an aircraft keeps it aloft with hundreds of people and tonnes of cargo on-board. Here is a short video of my flight taking off from Des Moines International Airport, Des Moines, Iowa.
Media 1: A commercial airliner taking off.
The video was recorded by the worst phone ever!
My phone is so bad that some of friends are trying
to start a fundraiser on Facebook to buy me a new phone!
The knowledge of the region surrounding a solid object immersed in a fluid is vital to the understanding of generation of lift and drag forces. How does an airfoil (a 2D cross section of a wing) generate lift? Do we have a complete understanding of the phenomenon? There are a number of popular approaches that try to explain aerodynamic lift but none of those provides a complete understanding of the phenomenon. Owing to the complexity of the flow around an airfoil, we do not know the exact causal chain of events that lead to the generation of lift. The strong coupling, interdependence and the cyclic-cause-and-effect nature of the local pressure and velocity around a lifting body renders it impossible to trace the origins of the lifting force. Being a student of fluid dynamics and turbulence, I took it upon myself this time to shed some light on this conundrum. In this essay, in addition to my own discussion and review, I will survey different theories that explain some aspects of the phenomenon of aerodynamic lift.
A. Setting up the ground
In nature, the blind and random processes of evolution and natural selection have solved the problem of mechanical flight for soaring birds millions of years ago, without any understanding of the physical principles involved. Why is it then impossible for us to come up with a simple theory on the origins of aerodynamic lift?
The problem is approached from two standpoints, one intuitive and another mathematical. Both of these approaches are complementary, not contradictory. The intuitive approach, as the name suggests, is intended to provide us with a non-technical explanation of the origin of lift, while the mathematical approach tries to quantitatively reduce the phenomenon to equations and their explanations. The two major theories that are invoked to explain the lift in qualitative terms are the Equal Transit Time Theory and the Skipping Stone Theory. Most of the modern encyclopedias, high school level textbooks and aviation magazines refer to these theories as the explanations of aerodynamic lift. However, both of these theories are incorrect and do not represent our complete understanding of the phenomenon of lift. The generation of lift finds its origins in the synergistic agency of fluid viscosity, vorticity and the effects of turning of flow.
B. Non-technical, qualitative theories
B.1 Equal Transit Time Theory
This is based on the assumption that the air molecules travelling along the top and the bottom surfaces of an airfoil meet at the trailing edge at the same time, hence the name. Since a typical airfoil has a longer contour on the top than the bottom, it follows that the airflow along the top surface must be faster than that along the bottom, in order to satisfy the requirement of the equal transit time. Experiments have shown that the speed of the flow along the top surface is much higher than what this theory demands; meaning the molecules moving along the top and bottom surfaces do not meet at the trailing edge of the airfoil. If we plug in the velocity obtained by assuming the equal transit time into the Bernoulli's Equation, the resulting pressure is under-predicted, which in turn yields low value of the lift force. This hints at the non-physicality of the assumptions made in this theory. Based upon the equal transit time assumption, it would not be possible for an airplane to fly inverted. However, airplanes (in air shows) and airfoils (in wind tunnels) are observed to be equally capable of generating lift. A simple lift calculation and a simple observation debunk this theory. This theory is, however, correct in stating that a pressure difference exists between the top and the bottom surfaces of an airfoil. Because the theory is based on a non-physical assumption of equal transit time, it is not able to correctly quantify that (pressure) difference. Refer to Figure 1 to locate the high pressure - low velocity and low pressure - high velocity regions below and above the airfoil. The streamlines indicate the direction of the incoming flow -- which approaches the airfoil at an angle of about \(8^{\circ}\).
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| Figure 1: Velocity and pressure contours around an airfoil. Flow is from left to right. Simulation run using SIMPLE algorithm, which is a steady-state solver for incompressible, turbulent flow. SIMPLE stands for Semi-Implicit Method for Pressure Linked Equations. Click to enlarge. |
B.1.1 A youthful folly: Einstein's excursions into aeronautics
Albert Einstein's genius and curiosity has had no limits. In addition to producing the works like the discovery of the Photoelectric Effect, and the General Theory of Relativity, he digressed into other fields of physics including Fluid Dynamics. In early 20th century, Einstein developed a theory for ideal fluids, similar to the Bernoulli's Principle. According to his theory, Einstein attributed the aerodynamic lift to the differences in and the non-complementary nature of pressure and velocity above and below an airfoil. Based on his theory, he designed an airfoil called cat's back airfoil which had a long, humped top surface, so as to allow a longer path for the flow to travel, resulting in high velocity and low pressure. For testing, he presented his designs to aircraft manufacturer LVG (Luftverkehrsgesellschaft) in Berlin. A test pilot who was in-charge of testing Einstein's cat's back airfoil reported that the craft waddled around in the air like “a pregnant duck.” In 1954, Einstein himself called his digression into aeronautics a youthful folly! It is quite astonishing that one of the greatest scientists of the 20th century could not come up with a comprehensive theory of aerodynamic lift.
B.2 Skipping Stone Theory
C. Mathematical approach
The mathematical treatment of aerodynamic lift deduced from the dynamics of the flow past an airfoil can be traced back to the Helmholtz's Theorem, which loosely states that in the absence of rotational external forces, a fluid that is initially irrotational remains irrotational.
Let us begin with the flow of a perfect fluid (inviscid i.e., frictionless) past an airfoil. As shown in Figure 2(a), for a perfect fluid, the flow past an airfoil is characterized by two stagnation points -- one at the leading edge and the other on the top surface near the aft end. Under the action of frictional forces, a fluid near a solid wall adheres to it through Van der Waals forces of attraction between the molecules of the fluid and the solid surface. Since we are considering a perfect fluid in this case, viscosity (friction) has no role to play. The streamline pattern in Figure 2(a) is consistent with the Helmoltz Theorem, i.e., the flow of a perfect fluid (irrotational by definition) past an airfoil continues to remain irrotational. The irrotational flow field that develops around the airfoil produces zero vorticity and zero lift force \((F^{'}_L=0)\) on the airfoil due to the equivalence of pressure on top and bottom. This is known as D'Alembert's paradox. For more details, please refer to my earlier technical essay, Paradoxes and potentialities.
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| Figure 2: Circulation theory of lift. Figure adapted from Ellington [1]. |
On the contrary, for a real fluid, viscosity plays an important role. Due to the action of viscous forces, for a real fluid, the streamline pattern (Figure 2(a)) at the beginning of the motion, appears to deform in such a way so as to translate the aft stagnation point towards the trailing edge. Upon reaching the trailing edge, a vortex is shed as shown in Figure 3. A vortex is simply defined as a swirling mass of fluid. The presence of swirl implies that the particles of fluid within the region of the vortex possess a definite amount of rotation. In the jargon of fluid dynamics, the rotational strength of a vortex is measured by a quantity called Aerodynamic Circulation,
\[ -\Gamma = \oint\limits_{C} \mathbf{u} \cdot d \mathbf{s} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -(1)\]
Remember, we started out with a real flow which had no inherent rotation to begin with. Also, in the beginning the streamline pattern of a real fluid around the airfoil resembled that of a perfect fluid, albeit for a very short, transient interval. Since we started out with an irrotational system, invoking Helmoltz's Theorem tells us that the total circulation (rotation) of the flow must be zero at all times. Since the shed vortex has a circulation of \(-\Gamma\), to cancel it out, there must exist an equal but opposite circulation, \(\Gamma\) somewhere else in the system. We know that the dot product of the orthogonal vectors is zero, hence the contributions to circulation from the parts of the airfoil where the dominating components of the local fluid velocity \(\mathbf{u}\) and the contour element \(d\mathbf{s}\) are perpendicular, vanish. However, as the flow is predominantly 2D, the local fluid velocity lies in the same plane as the contour of the airfoil, such that the local fluid velocity \(\mathbf{u}\) and the contour element \(d\mathbf{s}\), are coplanar, non-orthogonal vectors. This means the dot product is just the multiplication of the magnitude of the local velocity, contour element vectors and the cosine of the angle between them. The contour direction changes from positive to negative as we traverse from the top surface to the bottom surface of the airfoil. The top and the bottom surfaces yield circulation integral values that have opposite sign. Due to differences in lengths and the local fluid velocity along the top and bottom contours of the airfoil, the integral as described in Equation (1) is non-zero. The value of this integral represents the net effective circulation about the airfoil (total flow minus the horizontal flow). See Figure 3.
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| Figure 3: Circulation flow field, and a shed, starting vortex. |
Circulation flow field (Figure 2b) superposed on top the non-lifting flow field (Figure 2a) accounts for the situation shown in Figure 2c. In other words, it is due to circulation that the flow velocity is larger on the top of the airfoil than the bottom, since the circulation flow field and and local non-lifting/ambient flow field augment each other on the top surface. On the bottom surface, as the circulation flow field is opposite to the non-lifting/ambient flow field, the resulting local flow velocity is smaller. This suggests a pressure differential between the top and the bottom surfaces of the airfoil, in the spirit of the Bernoulli's Principle, with higher pressure along the low-speed side (bottom surface) and lower pressure along the high-speed side (top surface). The pressure/velocity difference between the top and the bottom surfaces generates lift. The relation between the circulation and the lifting force is shown in Figure 2c, \(F^{'}_L = \rho U_{\infty} \Gamma\). This completes the description of the theoretical origin of aerodynamic lift.
It is to be noted that circulation is an indispensable mathematical tool to understand the origins of lift. Whether there is rotation of fluid around the airfoil in physically realistic sense is not an experimental question. If we subtract the non-lifting, ambient flow from the local flow around the airfoil, we will end up with a small region of faster flow on the top of the airfoil going in \((+X)\) direction, and a small region of slower flow along the bottom of the airfoil going in the opposite \((-X)\) direction. This gives rise to net circulation, which accounts for the mathematical origin of aerodynamic lift.
Cite as
Bader, Shujaut H., “The enigma of flight: A short note on the mystery of aerodynamic lift and its competing theories." Backscatter, July 9, 2021. https://backscatterblog.blogspot.com/2021/07/the-enigma-of-flight_01397133208.html
References
- C. P. Ellington. The aerodynamics of hovering insect flight. IV. aerodynamic mechanisms. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 305(1122):79–113, Feb. 1984c.
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